Weighted Composition Operators Acting between Kind of Weighted Bergman-Type Spaces and the Bers-Type Space

Authors: Amnah E. Shammahy

Abstract:

In this paper, we study the boundedness and compactness of the weighted composition operator W_u,φ, which is induced by an holomorphic function u and holomorphic self-map φ, acting between the N_K-space and the Bers-type space H^∞_α on the unit disk.

Keywords: Weighted composition operators, NK-space, Bers-type space.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1091212

References:

[1] R. Aulakari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their charactrization in terms of Carleson measures, Rocky Mountain J. Math. 26 (1996), 485-506.
[2] A. Bernstein II, Analytic functions of Bounded mean oscillation , In aspects of contemporary complex Analysi, Academic Press London (1980), 3-36.
[3] B. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183-2169.
[4] A. El-Sayed Ahmed and M.A. Bakhit, Holomorphic NK and Bergman-type spaces, Birkhuser Series on Operator Theory: Advances and Applications (2009), BirkhuserVerlag Publisher Basel-Switzerland, 195 (2009), 121-138.
[5] A. El-Sayed Ahmed and M. A. Bakhit, Hadamard products and NK spaces, Mathematical and Computer Modeling, 51 (2010), 33-43.
[6] A. El-Sayed Ahmed and M.A. Bakhit, Composition operators acting betweensome weighted M¨obius invariant spaces, J. Ann. Funct. Anal. 2 (2011), 138-152.
[7] M. Lindstr¨om and N. Palmberg, Spectra of composition operators on BMOA, Integr. Equ. Oper. Theory, 53 (2005), 75- 86.
[8] N. Palmberg, Composition operators acting on Np - spaces, Bull. Belg. Math. Soc. Simon Stevin, 14 (2007), 545 - 554.
[9] J. Pau, Bounded M¨obius invariant QK spaces, J. Math. Anal. Appl. 338 (2008), 1029-1042.
[10] S.-i. Ueki, Weighted Composition operators acting between the Np-space and the weighted-type space H∞α , Indagationes Mathematicae 23 (2012), 243-255.
[11] R. Zhao, On α-Bloch functions and VMOA, Acta. Math. Sci. 3 (1996), 349-360.
[12] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York 1990.
[13] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), 1143-1177.